Universit¨at Ulm Abteilung Theoretische Informatik Leiter: Prof. Dr. Uwe Sch¨oning ON THE COMPLEXITY OF SOME PROBLEMS IN LINEAR ALGEBRA DISSERTATION zur Erlangung des Doktorgrades Dr. rer. nat. der Fakult¨at f¨ur Informatik der Universit¨at Ulm vorgelegt von THANH MINH HOANG Ulm, November 2003 Amtierender Dekan: Prof. Dr. F. W. von Henke Gutachter: Prof. Dr. Uwe Sch¨oning Prof. Dr. Thomas Thierauf Prof. Dr. Jacobo Tor´an Tag der Promotion: 19. Januar 2004 Acknowledgments Foremost, I would like to thank my supervisor, Thomas Thierauf. His continuous support and guidance have been greatly influenced my research and this work. I am grateful to Uwe Sch¨oning and Jacobo Tor´an for teaching me in theoretical computer science and for refereeing my thesis. I wish to thank Eric Allender and Meena Mahajan for many helpful comments and discussions on the topic of my thesis. Finally, I thank my parents, my family, Hang, Viet, and Duc for love and patience. Contents 1 Introduction 1 2 Preliminaries 15 2.1 Linearalgebra............................. 15 2.1.1 Basicnotations ........................ 15 2.1.2 The rank, the determinant, and the characteristic polynomial 17 2.1.3 The invariant factors and canonical forms . 19 2.1.4 Equivalence relations on matrices . 21 2.2 Complexitytheory .......................... 22 2.2.1 Logspacecountingclasses . 23 2.2.2 Reducibility and logspace counting hierarchies . ... 25 2.2.3 Characterizations of logspace counting classes . ... 28 3 On the Matrix Structure 33 3.1 The characteristic polynomial of a matrix . .. 33 3.1.1 Computing the characteristic polynomial . 33 3.1.2 Verifying the characteristic polynomial . 35 3.2 Theinvariantfactorsofamatrix . 40 3.2.1 Computing and verifying the minimal polynomial . 41 3.2.2 Theinvariantfactorsystemofamatrix . 47 3.2.3 Moreabouttheminimalpolynomial . 50 3.3 Similarity and diagonalizability of matrices . ..... 60 3.3.1 Testingsimilarity . 61 3.3.2 Testing diagonalizability . 62 4 The Inertia of a Matrix 65 4.1 Computing and verifying matrix inertia . 65 4.1.1 Routh-Hurwitz’sTheorem . 66 4.1.2 Upperbounds......................... 68 i CONTENTS 4.1.3 Lowerbounds ......................... 71 4.2 Stabilityofmatrices . .. .. 73 5 Further Results 79 5.1 Uniqueperfectmatching . 79 5.2 Conditions for collapsing the C=L-hierarchy . 84 5.3 Conditions for further collapse of PL ................ 87 Conclusions and Open Problems 91 Bibliography 93 Deutsche Zusammenfassung 103 ii Chapter 1 Introduction One of the main subjects of theoretical computer science is complexity theory which is more or less concerned with determining the intrinsic complexity of computational problems. A very important aim of complexity theory is to clas- sify computational problems into different complexity classes defined by various bounds and types of resources (usually time and space). For such classifica- tion, the difficulty of some given computational problems within a complexity class can be meaningfully compared by using the fundamental concept of reduc- tions. Thereby, in many instances the hardest of all computational problems in a complexity class can be identified, they are called the complete problems for the considered complexity class. Another obvious aim of complexity theory is to search for the interrelationship between various complexity classes. Linear algebra is one of the most known mathematical disciplines because of its rich theoretical foundations and its many useful applications to science and engineering. Solving systems of linear equations and computing determinants are two examples of fundamental problems in linear algebra that have been studied for a long time ago. Leibnitz found the formula for determinants in 1693, and in 1750 Cramer presented a method for solving systems of linear equations, which is today known as Cramer’s Rule (see [AM87]). This is the first foundation stone on the development of linear algebra and matrix theory. At the beginning of the evolution of digital computers, the matrix calculus has received very much attention. John von Neumann and Alan Turing were the world-famous pioneers of computer science. They introduced significant contributions to the develop- ment of computer linear algebra. In 1947, von Neumann and Goldstine [vNG47] investigated the effect of rounding errors on the solution of linear equations. One year later, Turing [Tur48] initiated a method for factoring a matrix to a product of a lower triangular matrix with an echelon matrix (the factorization is known 1 CHAPTER 1. INTRODUCTION as LUdecomposition). At present, computer linear algebra is broadly of interest. This is due to the fact that the field is now recognized as an absolutely essen- tial tool in many branches of computer applications that require computations which are lengthy and difficult to get right when done by hand, for example: in computer graphics, in geometric modeling, in robotics, etc. In the complexity-theoretic viewpoint, in particular with respect to parallel computations, computational problems in linear algebra provide a sure enough interesting topic. The present thesis focuses on this topic. The main goal of this thesis is to determine precisely the complexity of some fundamental problems in linear algebra. On the other hand, the interrelationship between complexity classes in which the considered problems are located will be clarified. The motivation for this thesis comes mainly from the purpose to understand the complexity of computational problems in linear algebra. Many tasks of linear algebra are recognized usually as elementary problems, but the precise complexity of them was not known for a long time ago. Computing the determinant is an ex- ample. There are polynomial-time algorithms for the determinant (see [vzGG99], Part I, Chapter 1, Section 5.5., Algorithm 5.10). With respect to parallel com- putations, the problem of computing determinants attracted a great attention. It was shown in [Ber84, BvzGH82, Chi85, Csa76] that the determinant is com- putable simultaneously in polylog-time by using a polynomial number of proces- sors. In particular, the result of Berkowitz [Ber84] showed that the problem of computing the determinant is solvable by uniform Boolean circuits of O(log2n)- depth and polynomial-size, i.e. the determinant is in the class NC2 (see Chapter 2 below for more detail on NC and its subclasses). Many computational problems in linear algebra are reducible in a natural way to computing determinants, and hence they are known to be in NC2. However, NC2 does not capture the ex- act complexity of linear-algebraic problems. Taking in consideration that the determinant is not known to be NC2-complete, it is natural to ask for which complexity class this problem is complete, and whether the complexity of other fundamental problems in linear algebra can be found. Counting problems and counting classes Counting problem is a type of computational problems, which is more difficult than decision problem and search problem. The major difference between these three types of computational problems can be explained as follows: a decision problem asks whether a solution exists, a search problem demands to compute 2 a solution, but a counting problem counts the number of all solutions. The perfect matching problem for graphs seems to be a good example: for a given graph G, the decision problem asks whether there is a perfect matching in G, the search problem demands to construct one of the perfect matchings in G (if one exists), and the counting version requires to compute the number of all perfect matchings in G. Although the perfect matching decision problem can be solved in deterministic polynomial-time [Edm65], counting the number of all perfect matchings in a graph is a very difficult problem for which maybe no polynomial-time algorithm can be developed. Just now is the question: How difficult the problem of counting the number of all perfect matchings in a graph will be stepped?. The two most known time-complexity classes are P and NP. P is the class of all decision problems solvable in deterministic polynomial-time, and NP is the class of all decision problems solvable in nondeterministic polynomial-time. A deterministic polynomial-time algorithm is usually called efficient. Therefore, one can intuitively say that P contains only efficient computational problems. In contrast to P, NP-complete problems are called intractable, since no polynomial- time algorithm for any of these problems is known today. A large number of computational problems were shown by Cook and Karp [Coo71, Kar72] to be NP- complete. Whether there is a polynomial-time algorithm for any NP-complete problem is a formulation (in the algorithmic viewpoint) for the number-one open question P =? NP in theoretical computer science. Other complexity classes beyond NP were (and are) widely of interest. One of them is the counting class #P which is extended from NP among the most natural way. In 1979, Valiant [Val79b, Val79c] initiated the study of the compu- tational complexity of counting problems. He introduced the counting class #P that counts the number of solutions of NP-problems, or equivalently, the number of all accepting computation paths of a nondeterministic Turing machine on an input. Computing the number of all truth assignments satisfying a given Boolean ex- pression is the counting version corresponding to the very popular NP-complete decision problem SAT. This counting problem is known to be complete for #P (the proof can be found in [Pap94], Chapter 18, page 442). More interest- ingly, there are #P-complete counting problems derived from problems solvable in polynomial-time. For example: one can decide in polynomial-time whether a bipartite graph has a perfect matching [Edm65], but the number of all perfect matchings in a bipartite graph is a #P-complete function [Val79b]. Moreover, since the number of all perfect matchings in a bipartite graph G is equal to the 3 CHAPTER 1. INTRODUCTION permanent of the adjacency matrix of G, the problem of computing the perma- nent of 0-1 matrices is also #P-complete.